cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A368329 The largest term of A054743 that divide n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 16, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 32, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 16, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 64, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 16, 81, 1, 1, 1, 1
Offset: 1

Views

Author

Amiram Eldar, Dec 21 2023

Keywords

Comments

First differs from A360540 at n = 27.
The largest divisor d of n such that e > p for all prime powers p^e in the prime factorization of d (i.e., e >= 1 and p^(e+1) does not divide d).

Links

Crossrefs

Cf. A054743, A034444, A207481, A327939, A360540, A368329, A368330, A368331, A368333.

Programs

  • Mathematica
    f[p_, e_] := If[e <= p, 1, p^e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] <= f[i,1], 1, f[i,1]^f[i,2]));}

Formula

Multiplicative with a(p^e) = 1 if e <= p, and a(p^e) = p^e if e > p.
A034444(a(n)) = A368330(n).
a(n) >= 1, with equality if and only if n is in A207481.
a(n) <= n, with equality if and only if n is in A054743.
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 - 1/p^(s-1) + 1/p^((p+2)*s-1) - 1/p^((p+2)*(s-1)+1) - 1/p^((p+1)*s) + 1/p^((p+1)*(s-1))).

A368335 The number of divisors of the largest term of A054744 that divides of n.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 4, 3, 1, 1, 1, 6, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 3, 1, 4, 1, 4, 1, 1, 1, 3, 1, 1, 1, 7, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 5, 5, 1, 1, 3, 1, 1, 1
Offset: 1

Views

Author

Amiram Eldar, Dec 21 2023

Keywords

Links

Crossrefs

Cf. A000005, A048103, A054744, A368331, A368332, A368333, A368334, A368336.

Programs

  • Mathematica
    f[p_, e_] := If[e < p, 1, e+1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] < f[i,1], 1, f[i,2]+1));}

Formula

Multiplicative with a(p^e) = 1 if e < p, and a(p^e) = e+1 if e >= p.
a(n) = A000005(A368333(n)).
a(n) >= 1, with equality if and only if n is in A048103.
a(n) <= A000005(n), with equality if and only if n is in A054744.
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - 1/p^s + 1/p^(p*s-1) + 1/p^((p+1)*s) - 1/p^((p+1)*s-1)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + (1 + (p-1)*p)/((p-1)*p^p)) = 1.98019019497523582894... .

A129252 Smallest prime factor p of n such that p^p is a divisor of n, a(n)=1 if no such factor exists.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2
Offset: 1

Views

Author

Reinhard Zumkeller, Apr 07 2007

Keywords

Examples

			For n = 108 = 2^2 * 3^3, it is 2 that is the smallest prime factor p satisfying p^p | 108, thus a(108) = 2.

Links

Crossrefs

Cf. A020639, A048103 (positions of 1's), A008578, A051674, A129251, A359550, A368333, A380528.
Differs from A327936 for the first time at n=108.

Programs

  • Mathematica
    Array[If[IntegerQ@ #, #, 1] &@ First@ SelectFirst[FactorInteger[#], #1 <= #2 & @@ # &] &, 120] (* Michael De Vlieger, Oct 01 2019 *)
  • PARI
    A129252(n) = { my(f = factor(n)); for(k=1, #f~, if(f[k, 2]>=f[k, 1], return(f[k, 1]))); (1); }; \\ Antti Karttunen, Oct 01 2019
  • PARI
    A129252(n) = { my(pp); forprime(p=2, , pp = p^p; if(!(n%pp), return(p)); if(pp > n, return(1))); }; \\ Antti Karttunen, Feb 09 2025

Formula

a(n) = 1 iff A129251(n) = 0.
a(A048103(n)) = 1.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 - 1/p^p) + Sum_{p prime} ((1/p^(p-1)) * Product_{primes q < p} (1-1/q^q)) = 1.30648526015949409005... . - Amiram Eldar, Nov 07 2022

Extensions

Data section extended to a(120) by Antti Karttunen, Oct 01 2019

A368332 The number of terms of A054744 that divide n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 2, 1, 3, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 4, 3, 1, 1, 2, 1, 1, 1
Offset: 1

Views

Author

Amiram Eldar, Dec 21 2023

Keywords

Comments

The number of divisors d of n such that e >= p for all prime powers p^e in the prime factorization of d (i.e., e >= 1 and p^(e+1) does not divide d).
The largest of these divisors is A368333(n).

Links

Crossrefs

Cf. A054744, A365632, A368328, A368333, A368334, A368335.

Programs

  • Mathematica
    f[p_, e_] := If[e < p, 1, e - p + 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] < f[i,1], 1, f[i,2] - f[i,1] + 2));}

Formula

Multiplicative with a(p^e) = 1 if e < p, and a(p^e) = e - p + 2 if e >= p.
a(n) >= 1, with equality if and only if n is in A048103.
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - 1/p^s + 1/p^(p*s)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + 1/((p-1)*p^(p-1))) = 1.58396891058853238595... .

A368334 The number of terms of A054744 that are unitary divisors of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1
Offset: 1

Views

Author

Amiram Eldar, Dec 21 2023

Keywords

Comments

First differs from A081117 at n = 28.
Also, the number of terms of A072873 that are unitary divisors of n.

Links

Crossrefs

Cf. A034444, A048103, A054744, A072873, A327939, A368330, A368332, A368333, A368335.
Cf. A081117.

Programs

  • Mathematica
    f[p_, e_] := If[e < p, 1, 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] < f[i,1], 1, 2));}

Formula

Multiplicative with a(p^e) = 1 if e < p, and a(p^e) = 2 if e >= p.
a(n) = A034444(A368333(n)).
a(n) = A034444(A327939(n)).
a(n) >= 1, with equality if and only if n is in A048103.
a(n) <= A034444(n), with equality if and only if n is in A054744.
Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 1/p^(p*s)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + 1/p^p) = 1.29671268566745796443... .
Showing 1-5 of 5 results.

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